Problem: Determine how many solutions exist for the system of equations. ${3x-y = 1}$ ${-3x+y = 3}$
Explanation: Convert both equations to slope-intercept form: ${3x-y = 1}$ $3x{-3x} - y = 1{-3x}$ $-y = 1-3x$ $y = -1+3x$ ${y = 3x-1}$ ${-3x+y = 3}$ $-3x{+3x} + y = 3{+3x}$ $y = 3+3x$ ${y = 3x+3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-1}$ ${y = 3x+3}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.